Theorem rexlimddvcbv 44991

Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44989. The equivalent of this theorem without the bound variable change is rexlimddv 3174. Usage of this theorem is discouraged because it depends on ax-13 2406, see rexlimddvcbvw 44990 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rexlimddvcbv.1	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃)
rexlimddvcbv.2	⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓)
rexlimddvcbv.3	⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒))

Assertion

Ref	Expression
rexlimddvcbv	⊢ (𝜑 → 𝜓)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑦   𝜒,𝑥   𝜃,𝑦   𝑥,𝐴   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)

Proof of Theorem rexlimddvcbv

Step	Hyp	Ref	Expression
1		rexlimddvcbv.1	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃)
2		rexlimddvcbv.3	. . 3 ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒))
3		rexlimddvcbv.2	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓)
4	2, 3	rexlimdvaacbv 44989	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜃 → 𝜓))
5	1, 4	mpd 16	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  ∃wrex 3091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092

This theorem is used by: (None)